Received September 21, 2012, in final form January 29, 2013; Published online February 05, 2013

Abstract
This paper is a continuation of the paper [De Bie H., Ørsted B., Somberg P.,
Souček V., Trans. Amer. Math. Soc.364 (2012), 3875–3902], investigating a natural radial
deformation of the Fourier transform in the setting of Clifford analysis.
At the same time, it gives
extensions of many results obtained in [Ben Saïd S., Kobayashi T., Ørsted B.,
Compos. Math.148 (2012), 1265–1336].
We establish the analogues of Bochner's formula and the Heisenberg uncertainty relation in the
framework of the (holomorphic) Hermite semigroup, and also give a detailed analytic treatment of
the series expansion of the associated integral transform.